The Switzerland midfielder completed his move from Bayern Munich on a four-and-a-half-year contract last Friday.
Shaqiri had previously been linked with transfers to Premier League sides Liverpool and Stoke City.
However, former Manchester City boss Mancini persuaded him to spurn the advances of other clubs and make the move to San Siro.
"I had other offers, but my choice was always clear," Shaqiri said. "The coach showed me his project, I was convinced by his words and now I want to help the club return to the highest levels.
"In spite of other offers, I always wanted Inter. I was very struck by the tifosi [fans] when I landed. It was a unique experience and I will never forget it.
"Taking great steps together with this coach and my team-mates is a beautiful experience."
The former Basel man also confirmed reports that Serie A leaders and defending champions Juventus were interested in signing him. But Shaqiri is now firmly focused on helping Inter challenge for the top three and a UEFA Champions League place.
"I'm hungry for victories and I want them to be with Inter," he added. "My first impression is excellent, I'm sure we'll go back to the highest levels.
"Reading the papers it looks like I've been in contact with a whole lot of teams.
"My brother [and agent] was at work and Juventus was one of the options too, but I only wanted Inter.
"I can't wait to play and I think third place is within our grasp. We picked up some points over the weekend and I'm sure we'll be able to fly high again."
Shaqiri was on the bench in Sunday's 3-1 win over Genoa and could make his debut this weekend as Inter head to Empoli.
